# Tire

Model tire dynamics and motion at end of driveline axis

## Library

Vehicle Components

## Description

The Tire block models a vehicle tire in contact with the road. The driveline port transfers the torque from the wheel axis to the tire. You must specify the vertical load Fz and vehicle longitudinal velocity Vx as Simulink® input signals. The model provides the tire angular velocity Ω and the longitudinal force Fx on the vehicle as Simulink output signals. All signals have MKS units.

The convention for the Fz signal is positive downward. If the vertical load Fz is zero or negative, the horizontal tire force Fx vanishes. In that case, the tire is just touching or has left the ground.

The longitudinal direction lies along the forward-backward axis of the vehicle. See Tire Model following for model details.

### Using Vehicle Component Blocks

Use the blocks of the Vehicle Components library as a starting point for vehicle modeling. To see how a Vehicle Component block models a driveline component, look under the block mask. The blocks of this library serve as suggestions for developing variant or entirely new models to simulate the same components. Break the block's library link before modifying it and creating your own version.

## Dialog Box and Parameters

Effective radius re, in meters (m), at which the longitudinal force is transferred to the wheel as a torque. The default is `0.3`.

Rated vertical load force Fz, in newtons (N). The default is `3000`.

Peak longitudinal force at rated load

Maximum longitudinal force Fx, in newtons (N), the tire exerts on the wheel when the vertical load equals its rated value. The default is `3500`.

Slip at peak force at rated load

The value of the contact slip, in percent (%), when Fx equals its maximum value and Fz equals its rated value. The default is `10`.

Tire relaxation length σκ, in meters (m), that determines the transient response of the tire force Fx at the tire's rated load. The default is `0.2`.

 Note:   For more about the tire parameters, see Relationship to Block Parameters following.

## Tire Model

The tire is a flexible body in contact with the road surface and subject to slip. When a torque is applied to the wheel axle, the tire deforms, pushes on the ground (while subject to contact friction), and transfers the resulting reaction (including rolling resistance) as a force back on the wheel, pushing the wheel forward or backward.

The Tire block models the tire as a rigid-wheel, flexible-body combination in contact with the road. The model includes only longitudinal motion and no camber, turning, or lateral motion.

• At full speed, the tire acts like a damper, and the longitudinal force Fx is determined mainly by the slip.

• At low speeds, when the tire is starting up from or slowing down to a stop, the tire behaves more like a deformable, circular spring.

• The effective rolling radius re is normally slightly less than the nominal tire radius because the tire deforms under its vertical load.

• The tire relaxation length σκ is the ratio of the slip stiffness to longitudinal force stiffness. It determines the transient response of Fx to slip.

This figure and table define the tire model variables. The figure displays the forces from the ground on the tire. The definition here differs from the normal convention in which Fz is positive downward, the force from the tire on the ground.

Tire Dynamics and Motion

Tire Model Variables and Constants

SymbolMeaning and Unit
IwWheel-tire assembly inertia (kg·m2)
τdriveTorque applied by the axle to the wheel (N·m)
VxWheel center longitudinal velocity (m/s)
Vsx = VxreΩWheel slip velocity (m/s)
V′sx = VxreΩ′Contact point slip velocity (m/s)
κ = –Vsx/|Vx|Wheel slip
κ′ = –Vsx/|Vx|Contact patch slip
uTire longitudinal compliance or deformation (m)
Fx = f(κ′, Fz)Longitudinal force exerted by the tire on the wheel at the contact point (N). Also a characteristic function f of the tire.
CFx = (∂Fx/∂u)0Tire longitudinal stiffness (N/m)
σκ = (∂Fx/∂κ′ )0/(CFx)Tire relaxation length (m)

### Tire Response

#### Roll and Slip

If the tire were rigid and did not slip, it would roll and translate as Vx = reΩ. In reality, even a rigid tire slips, and a tire develops a longitudinal force Fx only in response to slip. The wheel slip velocity Vsx = VxreΩ ≠ 0. The wheel slip κ = –Vsx/|Vx| is more convenient. For a locked, sliding tire, κ = –1. For perfect rolling, κ = 0.

#### Deformation

The tire is also flexible. Because it deforms, the contact point turns at a slightly different angular velocity Ω′ from the wheel. The contact point slip κ′ = –Vsx/|Vx|, where V′sx = VxreΩ′.

The tire deformation u directly measures the difference of wheel and contact point slip and satisfies

$\frac{du}{dt}={{V}^{\prime }}_{sx}-{V}_{sx}$

#### Forces and Characteristic Function

A tire model provides a steady-state tire characteristic function Fx = f(κ′, Fz), the longitudinal force Fx the tire exerts on the wheel given

• Contact slip κ′

The contact slip κ′ in turn depends on the deformation u. The longitudinal force Fx is approximately proportional to the vertical load because Fx is generated by contact friction and the normal force Fz. (The relationship is somewhat nonlinear because of tire deformation and slip.) The dependence of Fx on κ′ is more complex.

### Tire Dynamics

The tire model incorporates transient as well as steady-state behavior and is thus appropriate for starting from, and coming to, a stop.

Because a rolling, stressed tire is not in a steady state, the contact slip κ′ and deformation u are not constant. Before they can be used in the characteristic function, their time evolution must be accounted for. In this model, u and κ′ are moderate to small. The relationships of Fx to u and u to κ′ are then linear:

$\begin{array}{l}{F}_{x}={C}_{Fx}\cdot u={C}_{F\kappa }\cdot {\kappa }^{\prime },{C}_{Fx}={\left(\frac{\partial {F}_{x}}{\partial u}\right)}_{u=0},{C}_{F\kappa }={\left(\frac{\partial {F}_{x}}{\partial {\kappa }^{\prime }}\right)}_{{\kappa }^{\prime }=0}\\ u={\sigma }_{\kappa }\cdot {\kappa }^{\prime },{\sigma }_{\kappa }={\left(\frac{\partial {F}_{x}}{\partial {\kappa }^{\prime }}\right)}_{{\kappa }^{\prime }=0}/{\left(\frac{\partial {F}_{x}}{\partial u}\right)}_{u=0}={C}_{F\kappa }/{C}_{Fx}\end{array}$

These properties are taken from empirical tire data.

The deformation u evolves according to

$\frac{du}{dt}+\left(\frac{1}{\sigma \kappa }\right)\cdot \text{​}|{V}_{x}|u=-{V}_{sx}$

The slip κ′ follows from σκ and u. The tire behaves like a driven damper of damping rate |Vx|/σκ.

#### Tire Dynamics at Low Speeds

At low speeds, the slip remains finite, and the tire behaves more like a circular spring of stiffness CFx. In this limit, the linear approximation relating contact slip κ′ and deformation u becomes singular if damping is not explicitly included. This relationship can be modified:

${\kappa }^{\prime }=\frac{u}{\sigma \kappa }\to {\kappa }^{\prime }=\left(\frac{u}{\sigma \kappa }-\frac{{k}_{V,\text{low}}}{{C}_{F\kappa }}{V}_{sx}\right)$

where smooth transition from zero speed is provided by

${k}_{V,\text{low}}=\left\{\begin{array}{cc}\frac{1}{2}{k}_{V,\text{low}}\left(0\right)\left\{1+\mathrm{cos}\left(\pi \frac{|{V}_{x}|}{{V}_{\text{low}}}\right)\right\}& ,|{V}_{x}|\le {V}_{\text{low}}\\ 0& ,|{V}_{x}|>{V}_{\text{low}}\end{array}$

A nonsingular tire evolution valid at vanishing speeds is

$\begin{array}{l}\left(\frac{1}{{C}_{Fx}}\right)\cdot \frac{d{F}_{x}}{dt}+\text{​}|{V}_{x}|{\kappa }^{\prime }=-{V}_{sx},\\ \left(\frac{1}{{C}_{Fx}}\right)\cdot \frac{\partial {F}_{x}}{\partial {\kappa }^{\prime }}\frac{d{\kappa }^{\prime }}{dt}+\text{​}|{V}_{x}|{\kappa }^{\prime }=-{V}_{sx}-\left(\frac{1}{{C}_{Fx}}\right)\cdot \frac{\partial {F}_{x}}{\partial {F}_{z}}\frac{d{F}_{z}}{dt}\end{array}$

The second form explicitly shows the dependence on a varying vertical load Fz.

### Wheel and Vehicle Dynamics

With the tire characteristic function f(κ′, Fz), the vertical load Fz, and the evolved u and κ′, you can find the longitudinal force Fx and wheel velocity Ω. From these, the equations of motion determine the wheel angular motion (the angular velocity Ω) and longitudinal motion (the wheel center velocity Vx):

$\begin{array}{l}{I}_{w}\frac{d\Omega }{dt}={\tau }_{\text{drive}}-{r}_{e}{F}_{x}\\ m\frac{d{V}_{x}}{dt}={F}_{x}-\text{​}mg\cdot \mathrm{sin}\beta \end{array}$

where β is the slope of the incline upon which the vehicle is traveling (positive for uphill), and m and g are the wheel load mass and the gravitational acceleration, respectively. τdrive is the driveshaft torque applied to the wheel axis.

### Relationship to Block Parameters

The effective rolling radius is re. The rated load normalizes the tire characteristic function f(κ′, Fz), and the peak force, slip at peak force, and relaxation length fields determine the peak and slope of f(κ′, Fz) and thus CFx and σκ.

## Examples

The example model drive_4wd_dynamicsdrive_4wd_dynamics combines two differentials with four tire-wheel assemblies to model the contact of tires with the road and the longitudinal vehicle motion.

The example model drive_vehicledrive_vehicle models an entire one-wheel vehicle, including Tire and Longitudinal Vehicle Dynamics blocks.

## References

Centa, G., Motor Vehicle Dynamics: Modeling and Simulation, Singapore, World Scientific, 1997.

Pacejka, H. B., Tire and Vehicle Dynamics, Society of Automotive Engineers and Butterworth-Heinemann, Oxford, 2002.